Impressum Heyting.thy

(*<*)
theory Heyting
imports
  Closures
begin

(*>*)
section‹ Heyting algebras \label{sec:heyting_algebras} ›

text‹

  (complete) lattices are Heyting algebras. The following development is oriented towards
  the derived Heyting implication in a logical fashion. As there are no standard classes for
 -(complete-)lattices we simply work with complete lattices.

 :
 ▪ Heyting
 ▪*<*)
 ▪‹ -- properties

 ›

  heyting_algebra = complete_lattice +
 assumes inf_Sup_distrib1: "∧ Heyting algebras \label{sec:heyting_algebras} ›
begin

definition heyting :: "'a ==> 'a ==> 'a" (infixr ‹\⟶
 "x \⟶_H y = ⊔ z ≤ces aretng algebr hefollowig devement isoiented trds

  heg <>\open> 
 shows "z ≤⟶ z ⊓> y" (is "?lhs ⟷
 (rule iffI)
 from inf_Sup_distrib1 have "⊔ a ≤ x ≤add: SUP_le_if inf_comut
 \tem🍋‹
 show "?rhs ==>(imad: heyting_deupri.com
 

 

  \<openSign.mandatory_path "heyting"›Y::'a set. ∧ (⊔y∈ y)"

  heyting_algebra
 

  commute:
 shows "x ⊓ 'a ==> 'a"a" (infise arer noss and classes for
  (simp add: hyigi.omue)

 ―(coplete-)atticewe simply work with complete latt.
 ^bold>\>⟶

  curry_conv:
 shows "(x ⊓ y🚫Sign.mandatory_path "heyting"›
by (simp addf)(metisssoc

lemma swap
   uncurrycurry
by<> y\<longrightarrow>  x longrightarrowgrightarrowH y \<^sub>H z)"

lemma absorb:
  shows "y ⊓ (x \<longrightarrow>_\<longrightarrow>_\<longrightarrow> = complete_lattice
    and inf_Sup_distrib1∧sqinter<SqunionY = (⊔sqintery)"
by (simp_all add: curry inf_absorb1 ac_simps)

lemma detachment: (simp_all d curry inf_absorrb1sims)
  shows "x ⊓⟶H y) = x ⊓sis1
    and "(x H y) ⊓<> y")
proofbyis.socle
  showassumes x"
  then show ?thesis2 by (imp add: ac)
qed

lemma discharge:
  assumes "x' ≤⟶ x'" (is ?thesis2)
  shows " '<sqinter (x H y) = x' ⊓
    nd⟶ x' = y ⊓⟶ (y H z) ≤⟶_prry
proof -
  from  showesis1inf
  showmps P \<^sub "
qed

lemma trans:
  shows "(x \<longrightarrow> z = (x \<subH y) ⊓ (x \<^sub> )"
by (metis curry detacefi heyting :: "'f currydetachment e_infI2)

lemma:
  shows<> y'
byimprans

lemma
  hows P \<^sub>H Q"
by (simp add: curry)

lemmainfR:
 s "bold₌ ⊔
by (simp add: order_eq_iff curry uncurry)

lemma mono:
  assumes "x' ≤ x"
  assumesge F"
  shows "x \< x' H y'"
using ams by by (mei ury detachment1) uncurry inf_commutenf_absorb2 l_infI1)

lemma strengthen[strg]:
  assumes "st_ord (¬) G"
  assumes "st_ord F Y Y'"
 showst_ord F (X (X \^><longrightarrow>_<>\open> The Galo property for for 🚫H z"
usingmsF;simpono

lemma mono2monole z"
  assumes "monotone orda (≥l r_trans
  assumes "monotone orda (≤) G"
  shows
byp: monotoneImonotoneDsms

lemma
  assumes^bold>⟶_"
  ssumes y"
  shows "x ≤ xbold<longrightarrow>_H ⊤"
by (mes ascurryry ngeast order.rlordrrs

lemma botL:
  
by (smd_le)

lemmav
  shows "x Hy\top> ⟷ 
byshowsqunion (y ⊓> x <^bold<\^y⟷ y" is ""?lh⟷

lemma refl[simp]:
  shows "x \<longrightarrow>_H x = ⊤"
by (simp add: top_conv)

lemmatopL[simp]:
  shows "⊤⟶H x = x"
by (mesdtcmn()itoplf)

lemma topR[simp]:
  shows "x \<longrightarrow>java.lang.NullPointerException
by ( d:_onv

lemma K[simp_eq_iffnf_sup_distrib1
  shows⟶_\<longrightarrowH x) = ⊤"
by "x <> <> (SqunionY. x ⊓ y)

subclass distrib_lattice
proofcomment>‹‹Proposition~1.5.3› ›
     fromSup_distrib1. x \sqinter a ≤ y" b (simp add: SUP_le_iff inf)
    using commute by fastforce
  then have "  sqinter> (y ⊔ y) ⊔ z)" for x y z :: 'a
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  then show "x ⊔ z) = (x \>y) ⊓ z)" for x y z :: 'a
    by (rule distrib_imp1)
qed

lemma supL:
  shows "(x ⊔_ute
by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

subclass fastforceirreducible_on_defr_iffdistrib

lemma inf_Sup_distrib
  shows "x ⊓ ⊔Y = (⊔Y. x ⊓
    and "\Y ⊓
by (simp_all heytingclose

lemma inf_SUP_distrib:
  shows "x \sqinter (⊔I. Y i) = (⊔I. x ⊓
    context eyting_alg
by (simp_all add: inf_Sueg

end "X\Longrightarrow thesis1(esonF_lowerheytingjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75

lemma eq_boolean_implication: ― the implications coincide in 🍋le java.lang.NullPointerException
  fixes x :: "_::boolean_algebra"
  shows
by (simpx∈⟶_\<longrightarrowH (⊓X. Q x))" (is "?lhs = ?rhs")

lemmas simp_thms =
  heyting.botL
  L
  heyting.
  heyting.refl

lemma Sup_p"ting_algebra
  fixes x :: "_::heyting_algebra"
  showslongleftrightarrow Sup_irreducible
by (fastforcedefheyting
             rime_on_imp_Sup_irreducible_onjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

paragraphsimp .)

lemma
  fixes>∈ Q x) =  <
  shows "x ∈ X ==>
    mcontoS \le>) F"
     (Sqinter \^oldlongrightarrowsub Q x) ∩
    and "P x ⊓) G"
proof 
  show cont_inf1>) Sup (≤y. x ⊓
  then then mcont) Sup (≤x<>yfora
  inge)
java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 47
qed

lemma INFL:
  Q:eyting_algebra
  shows "(\>∈⟶<>) (<>∈X. P x) \<longrightarrow>_H Q" (islhs
proof(rule antisym)
  show
  show "?rhs ≤ closure_axioms_definf_monooer.efl)
qed

lemmasPL=eigINsmeri

lemma INFR:
  fixes es P :: "heyting_algebra
  <>x_\<longrightarrow>H (⊓ (simp order_eq_iff heytingnf
by (simp show
   (meson INFI INF_lowerSign.parent_path›

lemmas Inf_simps = ―
  Inf_inf
  inf_Inf
  INF_inf_const1
  INF_inf_const2
  heyting.INFL
  eyting

lemma SUPL_le:
  fixesby (metisinf
  shows
by simp

lemma
  fixes P : mono2mono
  shows>\inX P <^oldjava.lang.NullPointerException
by (simp add: SUPE UP_upper

lemmaf
  fixes Q :: "_::heyting_algebra"
  shows "(\Squnion\inX P x ⊓ Q) = (⊔x∈ Q"
by (simp add: heyting.inf_SUP_distrib(2))

lemmanf_SUP
  fixes_lgebra
  shows "(⊔ Hx = ⊥"
by (simp add: heyting.inf_SUP_distrib(1))

lemmass >‹
  sup_SUP
  SUP_sup
  heytinging.nf_SUPSUP
  heyting.SUP_inf

lemma mcont2mcont_inf[cont_intro]:
  ixes :: _\Rightarrow 'a::heyting_algebra"
  fixes G :: "_ \<Rightarrow> 'a::heyting_algebra"
  assumes "mcont luba orda Sup (\<le>) F"
  assumes "mcont luba orda Sup  <bold\<not>\<^sub>H(x \<squnion> y) = \<^bold>\not>^Hx \<sqinter> \<^bold>\<not>\<^sub>Hy"
  shows "mcont luba orda Sup (\<le>) (\<lambda>x. F x \<sqinter> G x)"
proof -
  have mcont_inf1: "mcont Sup (\<le>) SupdocomplementInfeudocomplementf
    by (auto intro!: contI mcontI monotoneI intro:_I2lipytingUP_distrib)
  then have mcont_inf2: "mcont Sup (\<le>) Sup (\<le>) (lambdax. x \<sqinter> y   :: "':eyting_algebrag_algebra"
    by"Rightarrow (_::heyting_algebra)"
  from assms mcont_inf1 mcont_inf2 show ?thesis
    by (best intro: ccpo.mcont2mcont'[OF complete_lattice_ccpo] ccpo.mcont_const[OF complete_lattice_ccpo])
qed

lemma closure_imp_distrib_le: \<comment>\<open> \<^citet>\<open>
  fixes P Q :: "_ :: heyting_algebra"
   cl "losure_axioms (\<>)cl"
  assumes cl_inf: "\<And>x y. cl x \<sqinter> cl y \<le> cl (x 
  shows "P \<^bold>\<longrightarrow>\<^sub>H Q \<le> cl P \<^bold>\<longrightarrow>\<^sub>H cl Q"
proof -
  roml have\<^bold\<longrightarrow>\<^sub>H Q) \<sqinter> cl P \<le> cl (P \<^bold\and"x\<old<<\<subH )\sqinter ' > x'" (is ?)
    by (metis (mono_tags) closure_axioms_def inf_monoderfl
  also have "\<dots> \<le> cl ((P \<^bold>\<longrightarrow>\<^sub>H Q) \<sqinter> P)"
    by (simp add: cl_inf)
  also from cl have "\<dots> \<le> cl Q"
    by (metis (mono_tags) closure_axioms_def order.ssumes\>downwards.closed"
  lly siss
    by (simp add: heyting
qed

setup \<open>Sign.parent_path\<close>


paragraph

definition pseudocomplement :: "'a:heyting_algebra'(\><bold\not\<^sub>H _\close [75]75 where
  "bold>\<not>\<^sub>Hx = x \<^bold>\<longrightarrow>\<^sub>H \<bottom>"

lemma pseudocomplementI
  shows "x \<le> \<^bold>\<not>\<^sub>Hy \<longleftrightarrow> x \<sqinter> y \<le> \<bottom>"
by (simp add: pseudocomplement_def heyting)

setup \open>Sign.mandatory_path "pseudocomplement"\<close>

lemma monotone:
  shows "antimonopseudocomplement"
by (simp add: antimonoI heyting.mono pseudocomplement_def)

lemmas strengthen[strg] = st_monotone[OF pseudocomplement.monotone]
lemmas mono = toneDeD[OF pseudocomplement.monotone]
lemmas mono2mono[cont_intro    IntD1ntD1ntD22ownwardsimp_mp'setI)java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
   = monotone2monotone[OF pseudocomplement.monotone, simplified _ y force dsmp_def

lemma eq_boolean_negation
  fixes  x::":{,heyting_algebra"
  shows "\<^bold>\<not>\<^sub>Hx = -x"
by (simp add: pseudocomplement_def heyting.eq_boolean_implication)

lemma  imp_boolean_implication_subseteq
  shows "x \<^bold>\<longrightarrow>\subH \<^bold>\<not>\<^sub>Hx = \<^bold>\<not>\<^sub>Hx"
by (simp add: pseudocomplement_defeq_iff_fytingting heytingngetachment

lemma:
  shows "x \< \<^bold>\<not>\<^sub>Hx = \<bottom>"
    and "\<^bold>\< botL
bylddseudocomplement_defeytingchment

lemmadouble_le:e_le
  shows" \<>\<^bold>\<not>\<^sub>H\<^bold>\<not>\<^sub>Hx"
byddpseudocomplement_deftachment ytingurry

interpretation doublee:closure_complete_lattice_class_attice_class "java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 0
by standard (simp; meson order.trans pseudocomplement.double_le pseudocomplement.mono)

lemma triple:
  shows "\<^bold>\<not>\<^sub>H\<^bold>\<not>\<^sub>H\<^bold>\<not>\<^sub>Hx = \<^bold>\\<^sub>Hx"
by (simp add: order_eq_iff pseudocomplement.double_le pseudocomplement.mono)

lemmacontrapos_le:
  shows "x \<^bold>\<longrightarrow>\<^sub>H y \<le> \<^bold>\<not>\<^sub>Hy \<^bold>\<longrightarrow>\<^sub>H \<^bold>\<not>\<^sub>Hx"
by (simp add: heyting.curry heyting.trans pseudocomplement_def)

lemma sup_inf: \<comment>\<open> half of de Morgan \<close>
  shows "\<^bold>\<not>\<^sub>H(x \<squnion> y) = \<^bold>\<not>\<^sub>Hx \<sqinter> \<^bold>\<not>\<^sub>Hy"
by (simp add: pseudocomplement_def heyting.supL)

lemma inf_sup_weak: \<comment>\<open> the weakened other half of de Morgan \<close>
  shows "\<^bold>\<not>\<^sub>H(x \<sqinter> y) = \<^bold>\<not>\<^sub>H\<^bold>\<not>\<^sub>H(\<^bold>\<not>\<^sub>Hx \<squnion> \<^bold>\<not>\<^sub>Hy)"
by (metis (no_types, opaque_lifting) pseudocomplement_def heyting.curry_conv heyting.supL inf_commute pseudocomplement.triple)                   

lemma fix_triv:
  assumes "x = \<^bold>\<not>\<^sub>Hx"
  shows "x = y"
using assms by (metis antisym bot.extremum inf.idem inf_le2 pseudocomplementI)

lemma double_top:
  shows "\<^bold>\<not>\<^sub>H\<^bold>\<not>\<^sub>H(x \<squnion> \<^bold>\<not>\<^sub>Hx) = \<top>"
by (metis pseudocomplement_def heyting.refl pseudocomplement.Inf(1) pseudocomplement.sup_inf)

lemma Inf_inf:
  fixes P :: "_ \<Rightarrow> (_::heyting_algebra)"
  shows "(\<Sqinter>x. P x) \<sqinter> \<^bold>\<not>\<^sub>HP x = \<bottom>"
by (simp add: pseudocomplement_def Inf_lower heyting.discharge(1))

lemma SUP_le: \<comment>\<open> half of de Morgan \<close>
  fixes P :: "_ \<Rightarrow> (_::heyting_algebra)"
  shows "(\<Squnion>x\<in>X. P x) \<le> \<^bold>\<not>\<^sub>H(\<Sqinter>x\<in>X. \<^bold>\<not>\<^sub>HP x)"
by (rule SUP_least) (meson INF_lower order.trans pseudocomplement.double_le pseudocomplement.mono)

lemma SUP_INF_le:
  fixes P :: "_ \<Rightarrow> (_::heyting_algebra)"
  shows "(\<Squnion>x\<in>X. \<^bold>\<not>\<^sub>HP x) \<le> \<^bold>\<not>\<^sub>H(\<Sqinter>x\<in>X. P x)"
by (simp add: INF_lower SUPE pseudocomplement.mono)

lemma SUP:
  fixes P :: "_ \<Rightarrow> (_::heyting_algebra)"
  shows "\<^bold>\<not>\<^sub>H(\<Squnion>x\<in>X. P x) = (\<Sqinter>x\<in>X. \<^bold>\<not>\<^sub>HP x)"
by (simp add: order.eq_iff SUP_upper le_INF_iff pseudocomplement.mono)
   (metis inf_commute pseudocomplement.SUP_le pseudocomplementI)

setup \<open>Sign.parent_path\<close>


subsection\<open> Downwardsclosure of  (downsets)\label{secclosures-} \<>

text\<open>

A \<^emph>\<open>downset\<close> (also \<^emph>\<open>lower set\<close> and \<^emph>\<open>order ideal\<close>) is a subset of a preorder that is closed under
the order relation. (An \<^emph>\<open>ideal\<close> is a downset that is \<^const>\<open>directed\<close>.) Some results require
antisymmetry (a partial order).

References:
 \<^item> \<^citet>\<open>"Vickers:1989"\<close>, early chapters.
 \<^item> \<^url>\<open>https://en.wikipedia.org/wiki/Alexandrov_topology\<close>
 \<^item> \<^citet>\<open>\<open>\S3\<close> in "AbadiPlotkin:1991"\<close>

\<close>

setup \<open>Sign.mandatory_path "downwards"\<close>

definition cl :: "'a::preorder set \<Rightarrow> 'a set" where
  "cl P = {x |x y. y \<in> P \<and> x \<le> y}"

setup \<openSignparent_path\closejava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

interpretation downwards: closure_powerset_distributive downwards.cl \<comment>\<open> On preorders \<close>
proof standard
  show "(P \<subseteq> downwards.cl Q) \<longleftrightarrow> (downwards.cl P \<subseteq> downwards.cl Q)" for P Q :: "'a set"
    unfolding downwards.cl_def by (auto dest: order_trans)
  show "downwards.cl (heyting.
    heyting.opR
qed

interpretation downwards: closure_powerset_distributive_anti_exchange "(downwards.cl::_::order set \<Rightarrow>  <> Sup_irreducible
  \<comment>\<open> On partial orders; see \<^citet>\<open>"PfaltzSlapal:2013"\<close> \<close>
by standard (unfold downwards.cl_def; blast intro: anti_exchangeI antisym)

setup \<open>Sign.mandatory_path "downwards"\<close>

lemma cl_empty:
  shows "downwards.cl {} = {}"
unfolding downwards.cl_def by simp

lemma closed_empty[ closed_empty[iff]:
   { <in ."
using downwards.cl_def by fastforce

lemma clI[intro]:
  assumes "y \<in> P"
  assumes "x \<le> y"
  shows "x \<in> downwards.cl P"
unfolding closure.closed_def downwards.cl_def using assms by blast

lemma clE:
  assumes "x \<in> downwards.cl P"
  obtains y where "y \<in> P" and "x \<le> y"
using assms unfolding downwards.cl_def by fast

lemma closed_in:
  assumes "x \<in> P"
  assumes "y \<le> x"
  assumes "P \<in> downwards.closed"
  shows "y \<in> P"
using assms unfolding downwards.cl_def downwards.closed_def by blast

lemma order_embedding: \<comment>\<open> On preorders; see \<^citet>\<open>\<open>\S1.35\<close> in "DaveyPriestley:2002"\<close> \<close>
  fixes x :: "_::preorder"
  shows "downwards.cl {x} \<subseteq> downwards.cl {y} \<longleftrightarrow> x \<le> y"
using downwards.cl by (blast elim: downwards.clE)

text\<open>

The lattice of downsets of a set \<open>X\<close> is always a \<^class>\<open>heyting_algebra\<close>.

References
 \<^item> \<^citet>\<open>\<open>\S7.5\<close> in "Ono:2019"\<close>; uses upsets, points to \<^citet>\<open>"Stone:1938"\<close> as the origin
 \<^item> \<^citet>\<open>\<open>\S2.2\<close> in "Esakia:2019"\<close>
 \<^item> \<^url>\<open>https://en.wikipedia.org/wiki/Intuitionistic_logic#Heyting_algebra_semantics\<close>

\<close>

definition\<>x\inX  \^><longrightarrow>\<^sub>H  x  ( \^><longrightarrow><subH(<>\inX. Q x)) is"lhs=?rhs)
  "imp P Q = {\<sigma>. \<forall>\<sigma>'\<le>\<sigma>. \<sigma>' \<in> P \<longrightarrow> \<sigma>' \<in> Q}"

lemma imp_refl:
  shows "downwards.imp P P = UNIV"
by (simp add: downwards.imp_def)

lemma imp_contained:
  assumes "P \<subseteq> Q"
  shows "downwards.imp P Q = UNIV"
unfolding downwards.imp_def using assms by fast

lemma heyting_imp:
  assumes "P \<n downwards."
  shows "P \<subseteq> downwards.imp Q R \<longleftrightarrow> P \<inter> Q \<subseteq> R"
using assms unfolding downwards.imp_def downwards.closed_def by blast

lemma imp_mp':
  assumes "\<sigma> \<in> downwards.imp P Q"
  assumes "\<sigma> \<in> P"
  shows "\<sigma> \<in> Q"
using assms by (simp add: downwards.imp_def)

lemma imp_mp:
  shows "P \<inter> downwards.imp P Q \<subseteq> Q"
    and "downwards.imp P Q \<inter> P \<subseteq> Q"
by (meson IntD1 IntD2 downwards.imp_mp' subsetI)+

lemma imp_contains:
  assumes "X \<subseteq> Q"
  assumes "X \<in> downwards.closed"
  shows "X \<subseteq> downwards.imp P Q"
using assms by (auto simp: downwards.imp_def elim: downwards.closed_in)

lemma imp_downwards:
  assumes "y \<in> downwards.imp P Q"
  assumes "x \<le> y"
  shows "x \<in> downwards.imp P Q"
using assms order_trans by (force simp: downwards.imp_def)

lemma closed_imp:
  shows "downwards.imp P Q \<in> downwards.closed"
by (meson downwards.clE downwards.closedI downwards.imp_downwards)

text\<open>

The set \<open>downwards.imp P
\fixesP : _:eyting_algebra
Note that ``kernel' isa choice  interiorfunction

\<close>

lemma imp_boolean_implication_subseteq:
  shows downwardsimpPQ\<subseteq P\<bold><longrightarrow\<sub> Q"
unfolding downwards.imp_def boolean_implication.set_alt_def by blast

lemma downwards_closed_imp_greatest:
  assumes "R \<subseteq> P \<^bold>\<longrightarrow>\<^sub>B Q"
  assumes "R \<in> downwards.closed"
  shows "R \<subseteq> downwards.imp P Q"
using assms unfolding boolean_implication.set_alt_def downwards.imp_def downwards.closed_def by blast

definition kernel :: "'a::order set \<Rightarrow> 'a set" where
  "kernel X = \<Squnion>{Q \<in> downwards.closed. Q \<subseteq> X}"

lemma kernel_def2:
  shows "downwards.kernel X = {\<sigma>. \<forall>\<sigma>'\<le>\<sigma>. \<sigma>' \<in> X}" (is "?lhs = ?rhs")
proof(rule antisym)
  show "?lhs \<subseteq> ?rhs"
    unfolding downwards.kernel_def using downwards.closed_conv by blast
next
  have "x \<in> ?lhs" if "x \<in> ?rhs" for x
    unfolding downwards.kernel_def using that
    by (auto elim: downwards.clE intro: exI[where x="downwards.cl {x}"])
  then show "?rhs \<subseteq> ?lhs" by blast
qed

lemma kernel_contractive:
  shows "downwards.kernel X \<subseteq> X"
unfolding downwards.kernel_def by blast

lemma kernel_idempotent:
  shows "downwards.kernel (downwards.kernel X) = downwards.kernel X"
unfolding downwards.kernel_def by blast

lemma kernel_monotone:
  showsdownwards.ernel
unfolding downwards.kernel_def by (rule by subst .commute rule mcont_inf1)

lemma closed_kernel_conv:
  shows "X \<in> downwards.closed \<longleftrightarrow> downwards.kernel X = X"
unfolding downwards.kernel_def2 downwards.closed_def by (blast elim: downwards.clE)

lemma closed_kernel:
  shows "downwards.kernel X \<in> downwards.closed"
by (simp add: downwards.closed_kernel_conv downwards.kernel_idempotent)

lemma kernel_cl:
  shows "downwards.kernel (downwards.cl X) = downwards.cl X"
using downwards.closed_kernel_conv by blast

lemma cl_kernel:
  shows "downwards.cl (downwards.kernel X) = downwards.kernel X"
by (simp flip: downwards.closed_conv add: downwards.closed_kernel)

lemma kernel_boolean_implication:
  fixes P :: "_::order"
  shows "downwards.kernel (P \<^bold>\<longrightarrow>\<^sub>B Q) = downwards.imp P Q"
unfolding downwards.kernel_def2 boolean_implication.set_alt_def downwards.imp_def by blast

setup \<open>Sign.parent_path\<close>
(*<*)

end
(*>*)